Subgroups — C₂²×C₃×C₄×#168 #14411164

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×#168	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×#168	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄×#168	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₄	1	2	yes	—
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄²×C₅	C₄	1	6	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	6	yes	—
9	C₂²×C₃×#168	C₄	1	16	yes	direct product factor
10	C₂²×C₃×#168	C₂²	1	12	yes	—
11	C₃×C₄×#168	C₂²	1	16	yes	direct product factor
12	C₂²×C₃×C₄×C₈	—	5	1	no	maximal, Hall
13	C₂²×C₄²×C₅	C₂×C₃	1	1	yes	characteristic
14	C₂³×#168	C₂×C₃	1	2	yes	—
15	C₂×C₄×#168	C₂×C₃	1	12	yes	direct product factor
16	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	C₈	1	16	yes	complemented
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	3	yes	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	8	yes	—
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	12	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
22	C₃×C₄²×C₅	C₂×C₄	1	12	yes	—
23	C₃×C₄²×C₅	C₂³	1	4	yes	—
24	C₂×C₃×#168	C₂×C₄	1	48	yes	direct product factor
25	C₂×C₃×#168	C₂³	1	8	yes	—
26	C₂³×C₃×C₈	—	5	2	no	—
27	C₂²×C₃×C₄²	D₅	1	1	yes	characteristic
28	C₂×C₃×C₄×C₈	—	5	12	no	—
29	C₂³×C₄×C₅	C₃×C₄	1	2	yes	—
30	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₃×C₄	1	6	yes	—
32	C₂×C₄²×C₅	C₂²×C₃	1	6	yes	—
33	C₂²×#168	C₃×C₄	1	16	yes	direct product factor
34	C₂²×#168	C₂²×C₃	1	12	yes	—
35	C₄×#168	C₂²×C₃	1	16	yes	direct product factor
36	C₂²×C₄×C₈	—	5	1	no	Sylow
37	C₂³×C₃×C₅	C₄²	1	4	yes	—
38	C₂³×C₃×C₅	C₂×C₈	1	8	yes	—
39	C₂³×C₃×C₅	C₂²×C₄	1	3	yes	—
40	C₂×C₃×C₄×C₅	C₄²	1	12	yes	—
41	C₂×C₃×C₄×C₅	C₂×C₈	1	48	yes	complemented
42	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
43	C₂×C₃×C₄×C₅	C₂²×C₄	1	12	yes	—
44	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
45	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
46	C₃×#168	C₂²×C₄	1	16	yes	direct product factor
47	C₂³×C₃×C₄	Dic₅	1	2	yes	—
48	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
49	C₂²×C₃×C₈	—	5	16	no	—
50	C₂²×C₃×C₈	—	5	12	no	—
51	C₂×C₃×C₄²	Dic₅	1	6	yes	—
52	C₂×C₃×C₄²	C₂×D₅	1	6	yes	—
53	C₃×C₄×C₈	—	5	16	no	—
54	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₃×C₈	1	16	yes	complemented
56	C₂²×C₄×C₅	C₂×C₃×C₄	1	8	yes	—
57	C₂²×C₄×C₅	C₂×C₃×C₄	1	3	yes	—
58	C₂²×C₄×C₅	C₂×C₃×C₄	1	12	yes	—
59	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
60	C₄²×C₅	C₂×C₃×C₄	1	12	yes	—
61	C₄²×C₅	C₂³×C₃	1	4	yes	—
62	C₂×#168	C₂×C₃×C₄	1	48	yes	direct product factor
63	C₂×#168	C₂³×C₃	1	8	yes	—
64	C₂³×C₈	—	5	2	no	—
65	C₂²×C₄²	C₃×D₅	1	1	yes	characteristic
66	C₂×C₄×C₈	—	5	12	no	—
67	C₂²×C₃×C₅	C₄×C₈	1	16	yes	complemented
68	C₂²×C₃×C₅	C₂×C₄²	1	6	yes	—
69	C₂²×C₃×C₅	C₂²×C₈	1	12	yes	—
70	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
71	C₃×C₄×C₅	C₂×C₄²	1	6	yes	—
72	C₃×C₄×C₅	C₂²×C₈	1	16	yes	complemented
73	C₃×C₄×C₅	C₂³×C₄	1	2	yes	—
74	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
75	C₂²×C₃×C₄	#168	1	16	yes	direct product factor
76	C₂²×C₃×C₄	C₄×D₅	1	8	yes	—
77	C₂²×C₃×C₄	C₂×Dic₅	1	12	yes	—
78	C₂²×C₃×C₄	C₂×Dic₅	1	3	yes	—
79	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
80	C₂×C₃×C₈	—	5	48	no	—
81	C₂×C₃×C₈	—	5	8	no	—
82	C₃×C₄²	C₂×Dic₅	1	12	yes	—
83	C₃×C₄²	C₂²×D₅	1	4	yes	—
84	C₂³×C₅	C₃×C₄²	1	4	yes	—
85	C₂³×C₅	C₂×C₃×C₈	1	8	yes	—
86	C₂³×C₅	C₂²×C₃×C₄	1	3	yes	—
87	C₂×C₄×C₅	C₃×C₄²	1	12	yes	—
88	C₂×C₄×C₅	C₂×C₃×C₈	1	48	yes	complemented
89	C₂×C₄×C₅	C₂²×C₃×C₄	1	12	yes	—
90	C₂×C₄×C₅	C₂²×C₃×C₄	1	3	yes	—
91	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
92	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
93	#168	C₂²×C₃×C₄	1	16	yes	direct product factor
94	C₂³×C₄	C₃×Dic₅	1	2	yes	—
95	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
96	C₂²×C₈	—	5	16	no	—
97	C₂²×C₈	—	5	12	no	—
98	C₂×C₄²	C₃×Dic₅	1	6	yes	—
99	C₂×C₄²	C₂×C₃×D₅	1	6	yes	—
100	C₄×C₈	—	5	16	no	—
101	C₂×C₃×C₅	C₂×C₄×C₈	1	12	yes	complemented
102	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
103	C₂×C₃×C₅	C₂³×C₈	1	2	yes	—
104	C₂³×C₃	C₂×#168	1	8	yes	—
105	C₂³×C₃	C₄×Dic₅	1	4	yes	—
106	C₂³×C₃	C₂²×Dic₅	1	3	yes	—
107	C₂×C₃×C₄	C₂×#168	1	48	yes	direct product factor
108	C₂×C₃×C₄	C₄×Dic₅	1	12	yes	—
109	C₂×C₃×C₄	C₂×C₄×D₅	1	12	yes	—
110	C₂×C₃×C₄	C₂²×Dic₅	1	8	yes	—
111	C₂×C₃×C₄	C₂²×Dic₅	1	3	yes	—
112	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
113	C₃×C₈	—	5	16	no	—
114	C₂²×C₅	C₃×C₄×C₈	1	16	yes	complemented
115	C₂²×C₅	C₂×C₃×C₄²	1	6	yes	—
116	C₂²×C₅	C₂²×C₃×C₈	1	12	yes	—
117	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
118	C₄×C₅	C₂×C₃×C₄²	1	6	yes	—
119	C₄×C₅	C₂²×C₃×C₈	1	16	yes	complemented
120	C₄×C₅	C₂³×C₃×C₄	1	2	yes	—
121	C₂⁴	C₂×C₃×Dic₅	1	1	yes	characteristic
122	C₂²×C₄	C₃×#168	1	16	yes	direct product factor
123	C₂²×C₄	C₃×C₄×D₅	1	8	yes	—
124	C₂²×C₄	C₂×C₃×Dic₅	1	12	yes	—
125	C₂²×C₄	C₂×C₃×Dic₅	1	3	yes	—
126	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
127	C₂×C₈	—	5	48	no	—
128	C₂×C₈	—	5	8	no	—
129	C₄²	C₂×C₃×Dic₅	1	12	yes	—
130	C₄²	C₂²×C₃×D₅	1	4	yes	—
131	C₃×C₅	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Hall
132	C₂²×C₃	C₄×#168	1	16	yes	direct product factor
133	C₂²×C₃	C₂²×#168	1	12	yes	—
134	C₂²×C₃	C₂×C₄×Dic₅	1	6	yes	—
135	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
136	C₃×C₄	C₂²×#168	1	16	yes	direct product factor
137	C₃×C₄	C₂×C₄×Dic₅	1	6	yes	—
138	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
139	C₂×C₅	C₂×C₃×C₄×C₈	1	12	yes	complemented
140	C₂×C₅	C₂²×C₃×C₄²	1	1	yes	characteristic
141	C₂×C₅	C₂³×C₃×C₈	1	2	yes	—
142	C₂³	C₂×C₃×#168	1	8	yes	—
143	C₂³	C₃×C₄×Dic₅	1	4	yes	—
144	C₂³	C₂²×C₃×Dic₅	1	3	yes	—
145	C₂×C₄	C₂×C₃×#168	1	48	yes	direct product factor
146	C₂×C₄	C₃×C₄×Dic₅	1	12	yes	—
147	C₂×C₄	C₂×C₃×C₄×D₅	1	12	yes	—
148	C₂×C₄	C₂²×C₃×Dic₅	1	8	yes	—
149	C₂×C₄	C₂²×C₃×Dic₅	1	3	yes	—
150	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
151	C₈	—	5	16	no	—
152	C₂×C₃	C₂×C₄×#168	1	12	yes	direct product factor
153	C₂×C₃	C₂³×#168	1	2	yes	—
154	C₂×C₃	C₂²×C₄×Dic₅	1	1	yes	characteristic
155	C₅	C₂²×C₃×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
156	C₂²	C₃×C₄×#168	1	16	yes	direct product factor
157	C₂²	C₂²×C₃×#168	1	12	yes	—
158	C₂²	C₂×C₃×C₄×Dic₅	1	6	yes	—
159	C₂²	C₂³×C₃×Dic₅	1	1	yes	characteristic
160	C₄	C₂²×C₃×#168	1	16	yes	direct product factor
161	C₄	C₂×C₃×C₄×Dic₅	1	6	yes	—
162	C₄	C₂²×C₃×C₄×D₅	1	2	yes	—
163	C₃	C₂²×C₄×#168	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	C₂×C₃×C₄×#168	1	12	yes	direct product factor
165	C₂	C₂³×C₃×#168	1	2	yes	—
166	C₂	C₂²×C₃×C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₄²	D₅
commutator, hypocenter	C₅	C₂²×C₃×C₄×C₈
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂²×C₃×C₄²×C₅	C₂
socle, prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁴×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₃×C₄×#168	C₂	direct (N × H)
4. C₂²×C₄×#168	C₃	direct (N × H)
9. C₂²×C₃×#168	C₄	direct (N × H)
11. C₃×C₄×#168	C₂²	direct (N × H)
15. C₂×C₄×#168	C₂×C₃	direct (N × H)
17. C₂²×C₃×C₄×C₅	C₈	semidirect
24. C₂×C₃×#168	C₂×C₄	direct (N × H)
33. C₂²×#168	C₃×C₄	direct (N × H)
35. C₄×#168	C₂²×C₃	direct (N × H)
41. C₂×C₃×C₄×C₅	C₂×C₈	semidirect
46. C₃×#168	C₂²×C₄	direct (N × H)
55. C₂²×C₄×C₅	C₃×C₈	semidirect
62. C₂×#168	C₂×C₃×C₄	direct (N × H)
67. C₂²×C₃×C₅	C₄×C₈	semidirect
72. C₃×C₄×C₅	C₂²×C₈	semidirect
75. C₂²×C₃×C₄	#168	direct (N × H)
88. C₂×C₄×C₅	C₂×C₃×C₈	semidirect
93. #168	C₂²×C₃×C₄	direct (N × H)
101. C₂×C₃×C₅	C₂×C₄×C₈	semidirect
107. C₂×C₃×C₄	C₂×#168	direct (N × H)
114. C₂²×C₅	C₃×C₄×C₈	semidirect
119. C₄×C₅	C₂²×C₃×C₈	semidirect
122. C₂²×C₄	C₃×#168	direct (N × H)
131. C₃×C₅	C₂²×C₄×C₈	semidirect
132. C₂²×C₃	C₄×#168	direct (N × H)
136. C₃×C₄	C₂²×#168	direct (N × H)
139. C₂×C₅	C₂×C₃×C₄×C₈	semidirect
145. C₂×C₄	C₂×C₃×#168	direct (N × H)
152. C₂×C₃	C₂×C₄×#168	direct (N × H)
155. C₅	C₂²×C₃×C₄×C₈	semidirect
156. C₂²	C₃×C₄×#168	direct (N × H)
160. C₄	C₂²×C₃×#168	direct (N × H)
163. C₃	C₂²×C₄×#168	direct (N × H)
164. C₂	C₂×C₃×C₄×#168	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₄²×C₅	2
C₂³×C₃×#168	2
C₂×C₃×C₄×#168	2
C₂²×C₄×#168	3
C₂²×C₃×C₄×C₈	5

Subgroups — C22×C3×C4×#168 #14411164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×#168 #14411164